The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data

Let Omega subset of R-2 be a bounded smooth domain, we study the following anisotropic elliptic problem {-del(a(x)del u) + a(x)u = 0 in Omega, partial derivative u/partial derivative v = e(u) - s phi(1) - h(x) on partial derivative Omega, where v denotes the outer unit normal vector to partial derivative Omega, h is an element of C-0(,alpha)(partial derivative Omega), s > 0 is a large parameter, a(x) is a positive smooth function and phi(1) is a positive first Steklov eigenfunction. We show that this problem has an unbounded number of solutions for all sufficiently large s, which give a positive answer to a generalization of the Lazer-McKenna conjecture for this case. Moreover, the solutions found exhibit multiple concentration behavior around boundary maxima of a(x)phi(1) as s -> +infinity.